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ABSTRACT. It is now possible to calculate the K"-theory of a large class of singular curves over fields of characteristic zero. Roughly speaking, the ÜT-theory of a curve is the X-theory of its (smooth) normalization plus a few shifted copies of the if-theory of the field plus a "nil part." The nil part is a vector space depending only on the analytic type of the singularities, and may be computed locally. We completely compute the nil part for seminormal curves and give a conjectural calculation in general which depends upon cyclic homology.Until recently, very little has been known about the higher algebraic Ktheory of anything but finite fields. In this note we announce the computability of the X-theory of singular curves in characteristic zero in terms of the K-theory of smooth curves and fields. If the curve is seminormal, we give a complete calculation; otherwise, the calculation depends on the validity of: CONJECTURE. Let B be a finite integral extension of a ring A, and let I be the conductor ideal. Assume A contains Q, the rational numbers. Then

In this paper we introduce and study a conjecture, which we call the Artinian Berger Conjecture (or "ABC" 1 ), about the Kähler differentials Ω A/k of a finite dimensional commutative algebra A over a perfect field k. When char(k) = 0, the conjecture says this: if A is a subalgebra of a principal ideal algebra B, and Ω A/k injects into Ω B/k , then A is a principal ideal algebra. Here a principal ideal algebra is a finite dimensional commutative k-algebra so that every ideal is principal, i.e., of the form (x) for some x. To state the conjecture when char(k) = 0, we replace 'principal ideal algebra' by 'tame principal ideal algebra'; we will of course define 'tame' and restate the conjecture below.As the name suggests, this is an Artinian version of a conjecture formulated over 30 years ago by R. Berger in [B]. Berger's Conjecture concerns the coordinate ring R of a reduced curve over a perfect field k, and says that Ω R/k is torsion-free if and only if R is regular. (One direction is classical: if R is regular then Ω R/k is torsion-free, because it is a projective R-module [W, 9.3.14].) Main Theorem 0.1 ("ABC ⇒ BC"). If char(k) = 0, the Artinian Berger Conjecture implies Berger's Conjecture.To see the connection between the two conjectures, let R be the coordinate ring of a singular curve over k with integral closure S and total ring of fractions F . Since Ω S/k is torsion-free, it injects into Ω F/k = F ⊗ Ω R/k .

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